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International Journal of Modern Physics B

APPLICATION OF GENERALIZED FEYNMAN–HELLMANN THEOREM IN QUANTIZATION OF LC CIRCUIT IN THERMO BATH

Hong-Yi Fan, Xubing Tang

Published Oct 30, 2010• 0 citations
Publisher

Abstract

For the quantized LC electric circuit, when taking the Joule thermal effect into account, we think that physical observables should be evaluated in the context of ensemble average. We then use the generalized Feynman–Hellmann theorem for ensemble average to calculate them, which seems convenient. Fluctuation of observables in various LC electric circuits in the presence of thermo bath growing with temperature is exhibited.

DOI
10.1142/s0217979210054609

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For the quantized LC electric circuit, when taking the Joule thermal effect into account, we think that physical observables should be evaluated in the context of ensemble average. We then use the generalized Feynman–Hellmann theorem for ensemble average to calculate them, which seems convenient. Fluctuation of observables in various LC electric circuits in the presence of thermo bath growing with temperature is exhibited.
The Hellmann-Feynman theorem in a pure ensemble is extended to a quasidegenerate mixed ensemble (canonical) as an ensemble average value. The theorem is also extended to a grand canonical ensemble wherein average particle number is conserved. Connections with the thermodynamic observables are made. An approximate relation between $\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{C}∕\ensuremath{\partial}\ensuremath{\lambda}$ and $\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{G}∕\ensuremath{\partial}\ensuremath{\lambda}$ is established at the leading order of the saddle-point approximation.
We present a simple derivation of the Hellmann–Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples, which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model, and the Lipkin model. We show that the Hellmann–Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free energy is available, but there is no direct access to the thermal average of the operators themselves.